In their seminal paper Moseley, Niaparast, and Ravi introduced the Joint Replenishment Problem (JRP) with holding and backlog costs that models the trade-off between ordering costs, holding costs, and backlog costs in supply chain planning systems. Their model generalized the classical the make-to-order version as well make-to-stock version. For the case where holding costs function of all items are the same and all backlog costs are the same, they provide a constant competitive algorithm, leaving designing a constant competitive algorithm for arbitrary functions open. Moreover, they noticed that their algorithm does not work for arbitrary (request dependent) holding costs and backlog costs functions. We resolve their open problem and design a constant competitive algorithm that works for arbitrary request dependent functions. Specifically, we establish a 4-competitive algorithm for the single-item case and a 16-competitive for the general (multi-item) version. The algorithm of Moseley, Niaparast, and Ravi is based on fixed priority on the requests to items, and request to an item are always served by order of deadlines. In contrast, we design an algorithm with dynamic priority over the requests such that instead of servicing a prefix by deadline of requests, we may need to service a general subset of the requests.